Summation Equation, Trying to solve this recurrence forumla.

Hello. I've searched around a bit for a math forum where I could get help with this and this seems like the one I found where I could get some help with this. I was posed the following problem. Now I must admit it is over my head (as is most of the math on this forum) I was hoping that someone here could help give me an answer for this person. Or if not an answer, a reason why his probem makes no sense.

I want to predict the number of cups to be hit in beer pong each round, based upon shot percentage.

It's not just number of shots * percentage, because if you make 2 shots in a row, you get an extra turn. Even your extra turns can get extra turns..

Therefore, Number of cups per round is the recursive formula

C(t) = C(a^2 * t) + 2at where 0 < a < 1 for accuracy

So if you want to find the number of cups in 1 round, calculate c(1)
This can actually be reduced to the summation equation

for i = 0 to infinity -> 2a^(2i+1)

How do I solve this summation and get a formula?
I'd really appreciate it,